cp's OEIS Frontend

A223906 Poly-Cauchy numbers of the second kind -hat c_3^(-n).

%I A223906 #29 Sep 02 2025 21:05:45
%S A223906 17,51,161,531,1817,6411,23201,85731,322217,1227771,4729841,18379731,
%T A223906 71908217,282817131,1116854081,4424238531,17567551817,69882262491,
%U A223906 278365739921,1109974078131,4429431765017,17686337611851,70651190491361,282322298874531
%N A223906 Poly-Cauchy numbers of the second kind -hat c_3^(-n).
%C A223906 The poly-Cauchy numbers of the second kind hat c_n^k can be expressed in terms of the (unsigned) Stirling numbers of the first kind: hat c_n^(k) = (-1)^n*sum(abs(stirling1(n,m))/(m+1)^k, m=0..n).
%H A223906 Vincenzo Librandi, <a href="/A223906/b223906.txt">Table of n, a(n) for n = 1..1000</a>
%H A223906 Takao Komatsu, <a href="http://link.springer.com/article/10.1007/s11139-012-9452-0">Poly-Cauchy numbers with a q parameter</a>, Ramanujan J. 31 (2013), 353-371.
%H A223906 Takao Komatsu, <a href="http://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/1806-06.pdf">Poly-Cauchy numbers</a>, RIMS Kokyuroku 1806 (2012), p. 42-53.
%H A223906 Takao Komatsu, <a href="http://doi.org/10.2206/kyushujm.67.143">Poly-Cauchy numbers</a>, Kyushu J. Math. 67 (2013), 143-153.
%H A223906 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (9,-26,24).
%F A223906 Conjecture: a(n) = 2^(1+n)+3^(1+n)+4^n. G.f.: -x*(144*x^2-102*x+17) / ((2*x-1)*(3*x-1)*(4*x-1)). - _Colin Barker_, Mar 31 2013
%t A223906 Table[-Sum[StirlingS1[3, k] (-1)^k (k + 1)^n, {k, 0, 3}], {n, 30}]
%o A223906 (PARI) a(n) = -sum(k=0, 3, (-1)^k*stirling(3, k, 1)*(k+1)^n); \\ _Michel Marcus_, Nov 14 2015
%Y A223906 Cf. A223173.
%K A223906 nonn,easy,changed
%O A223906 1,1
%A A223906 _Takao Komatsu_, Mar 29 2013